Abstract
In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form H 5/Γ where Γ is a torsion-free subgroup of minimal index of the congruence two subgroup Γ 5 2 of the group Γ 5 of positive units of the Lorentzian quadratic form x 2/1 +... +x 5/2 -x 6/2. We also show that Γ 5 2 is a reflection group with respect to a 5-dimensional right-angled convex polytope in H 5. As an application, we construct a hyperbolic 5-manifold of smallest known volume 7ζ (3)/4.
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